Question

The tangent at the point P(x₁, y₁) to the parabola y² = 4ax meets the parabola y² = 4a(x + b) at Q and R, the coordinates of the mid-point of QR are

• A. (x₁ – a, y₁ + b)
• B. (x₁, y₁)
• C. (x₁ + b, y₁ + a)
• D. (x₁ – b, y₁ – b)

Answer 1

Answer: (B)

(x₁, y₁)

Answer 2

Equation of the tangents at P(x₁, y₁) to the parabola

y² = 4ax is yy₁ = 2a(x + x₁)

or 2ax – y₁y + 2ax₁ = 0 ….(i)

If M(h, k) is the mid-point of QR, then equation of QR a chord of the parabola y² = 4a(x + b) in term of its mid-point is ky – 2a (x + h) – 4ab

= k² – 4a (h + b) (using T = S ¢ )

or 2ax – ky + k² – 2ah = 0 …...(ii)

Since (i) and (ii) represent the same line, we have

$\frac{2a}{2a}$ = $\frac{y_{1}}{k}$ = $\frac{2ax_{1}}{k^{2} - 2ah}$

̃ k = y₁ and k² – 2ah = 2ax₁

̃ $y_{1}^{2}$ – 2ah = 2ax₁ ̃ 4ax₁ – 2ax₁ = 2ah

̃ h = x₁